# A problem regarding Convex Hull.

We have a Convex hull of a set $$X\subseteq R^{n}$$ defined as $$C$$, we need to prove that $$C$$ can we written as the following: $$\bar{C}=\sum_{i=1}^mt_ix_i$$

where $$m\geq 1,t_i\geq0, x_1,x_2,....,x_m\in X$$ and $$\sum_{i=1}^mt_i=1$$.

So, we need to prove that $$C=\bar{C}$$.

We do that by proving $$\bar{C}\subseteq C$$ and $$C\subseteq \bar{C}$$.

I understood the first part, but got stuck at the second one.

So, in order to prove $$C\subseteq \bar{C}$$, the proof says, we need to prove that $$\bar{C}$$ is convex.

That actually makes sense, since we know $$C$$ is the convex hull and a convex hull is the intersection of all convex sets that contain $$X$$. That is, $$C$$ is contained in all convex sets that contain $$X$$.

But I don't see how $$\bar{C}$$ could contain $$X$$. If we take $$m=1$$, it could only contain a subset $$\{x_1,x_2,.....,x_m\}$$ of $$X$$ and not the whole set.

Am I missing something ? Kindly help !

• In this sort of problem, you usually have either $X=\{x_1,\ldots,x_m\}$, or $\bar C = \{ \sum_{i=1}^m t_ix_i \mid m\ge 1,\ t_i\ge 0,\ x_i\in X,\ \sum_{i=1}^m t_i=1\}$. If neither of those apply, you can very well have an extremal point of $C$ that doesn't belong to $\bar C$, as you pointed out. Edit: what I meant in the second case is that $m\ge 1$ can be freely chosen in $\mathbb N$, which makes $\bar C$ the set of all convex combinations of $X$. – N.Bach Oct 19 '18 at 22:26
• The $x_k$ are not a fixed collection of points in $X$. They are an arbitrary selection of points in $X$. So $X \subset \overline{C}$ is immediate. (Aside, the notation conflicts with the usual notation for set closure.) – copper.hat Oct 19 '18 at 22:58

$$\overset {-} C$$ contains all points of the form $$\sum_{i=1}^{m} t_i x_i$$ with $$m \geq 1, t_i \geq o, x_i \in X$$. All quantities here are variables, nothing is fixed. It is obvious that every $$x \in X$$ can be written in above form by taking $$m=1,x_1=x,t_1=1$$ so $$\overset {-} C$$ contains every point of $$X$$. ($$x_1,x_2,..,x_m$$ are not fixed elements of $$X$$).